Which of the following options satisfies y(1)?

Question

Let A(x) and B(x) be functions defined by

A(x) = integral (exp(t^2-2t) dt) [upper bound:x lower bound: 0]

B(x) = integral (exp(t^2+2t) dt) [upper bound: x lower bound: 0]

where t is a dummy variable. Find y(1) and y(x) satisfies d^2y/dx^2 - 4y = exp(x^2), y(0) = y'(0) = 0.

Option 1: 1/4 [B(1)exp(2) - exp(-2)A(1)]

Option 2: 1/4[A(1)exp(2) - exp(-2)B(1)]

Option 3: 1/2[A(1)exp(2) - exp(-2)B(1)]

Option 4: 1/4[A(1)exp(-2) - exp(2)B(1)]

Option 5: None of the above

Which of the above is correct?

Answer 1

Hi Cheryl,

It's difficult to follow the text. Did you mean "where" instead of "and" in the phrase "y(1) and y(x)"? If so, the simplest way to solve might be to assume that y(x) = f(x)*A(x) + g(x)*B(x), and crank through the differential equation. The symmetries of the second derivative forms should clue you in to the proper choice. I'm also suspicious as to why the exp terms and the A / B terms exchange positions across the option expressions -- are these simple multiplications, or am I missing some implicit parentheses?

-- Cheers, -- Mr. d.